You can stop inappropriate loudness of the dominent hand, by concious thought. After a piece is learned, accentuating the correct part (usually the right top note) becomes part of your standard performance for that piece. A fully learned piece, you can concentrate on the emotion of the performance and leave the mechanics (including top note emphasis) to your lower brain. The mechanics of the piano don't matter (if it will do a consistent pp, most that I've played will), you should control with your ears and brain which line of notes is the loudest.
I was so left hand dominent, Mother assigned me piano lessons as a sort of home physical therapy. I had injured a finger of the right hand with a folding chair age three, and hardly ever used it. Mother set me to twenty reps of the Schmitt exercises (published by Schirmer) which emphasize strength building and independence of every finger. These were very useful to gaining control of my injured third finger. (the pad is scar tissue below the nail) I also progressed through the usual John Thompson methods from pre-A through book one.
Then the professional teacher Mother hired moved me from Schmitt to the Edna Mae Berman excercise books which I learned from book one through book four. I then progressed on to Czerny exercises. I was also doing more fun performance pieces of course, but "make the melody sing" was a standard instruction that kept me concious of my right hand. I didn't need a lot of telling, it was a goal I agreed with however unnatural it was.
The teacher did allow me to play pieces where the left hand makes a significant contribution from time to time. Lecuona's Malaguena was a lot of fun with the crashing four octave run near the end.
I continue to have my own specialties. In my goal to "finish" Beethoven's Moonlight Sonata as an adult, in the third movement are passages where there is one melody and two countermelodies going at once. In my interpretation, each repetition of this section, a different melody or counter melody dominates. So occasionally and in turn, one or the other left hand counter melody dominates. This is an interpretation that you never hear on recordings, and I'm sure it would disqualify me from winning any contest, but I like it. Music is an art the expresses emotion, I can occasionally celebrate my difference.
Meanwhile, piano has helped me cope with right handed can openers, scissors, doorknobs, zippers, electronic enter key, in a righty dominated world.
Hope you have fun with your piano adventure.